Part 7 Optimization in Functional Space 7.0 Motivation Example

Maximizing Yield of Batch Reaction

Maximizing Yield of Batch Reaction

Part 7 Optimization in Functional Space 7.1 Calculus of Variation

Objective Functions

Equivalence of Lagrange and Bolza Forms

Equivalence of Bolza and Mayer Forms

Example

Problem Statement

Intuitive Interpretation Let’s visualize a competition, to which only functions which have 2 derivatives in (a,b) and which take on the prescribed end values are permissible. Let’s further assume that there exists a x*(t) such that I is the smallest.

Variation

Necessary Condition

Integration by Parts - 2nd Term

Euler-Lagrange Equation

Example

Transversality Conditions

Transversality Conditions

Example 1

Solution of Example 1

Example 2

Solution of Example 2

Dependent Boundary Conditions

Dependent Boundary Conditions

Unspecified Terminal Time We now consider a generalized problem where the final time is defined as the first time after the initial time t0 that the state trajectory is a member of a target set or terminal manifold.

Problem Definition

Variations of Optimal Trajectory and Terminal Time

Necessary Conditions

Terminal Constraint

Necessary Conditions

Example

Example

Vector Formulation

Example

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